# Properties

 Label 18T88 Order $$162$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_3\wr C_3:C_2$

# Related objects

## Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $88$ Group : $C_3\wr C_3:C_2$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,4,17)(2,3,18), (1,8,14)(2,7,13)(3,10,16)(4,9,15)(5,11,17)(6,12,18), (1,3)(2,4)(5,12)(6,11)(7,15)(8,16)(9,13)(10,14)(17,18) $|\Aut(F/K)|$: $6$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
18:  $C_3^2:C_2$
54:  $(C_3^2:C_3):C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 9: $(C_3^3:C_3):C_2$

## Low degree siblings

9T21 x 3, 18T88 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $6$ $3$ $(11,14,15)(12,13,16)$ $3, 3, 3, 3, 1, 1, 1, 1, 1, 1$ $6$ $3$ $( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$ $3, 3, 3, 3, 1, 1, 1, 1, 1, 1$ $3$ $3$ $( 5, 8, 9)( 6, 7,10)(11,15,14)(12,16,13)$ $3, 3, 3, 3, 1, 1, 1, 1, 1, 1$ $3$ $3$ $( 5, 9, 8)( 6,10, 7)(11,14,15)(12,13,16)$ $2, 2, 2, 2, 2, 2, 2, 2, 2$ $27$ $2$ $( 1, 2)( 3,17)( 4,18)( 5,12)( 6,11)( 7,15)( 8,16)( 9,13)(10,14)$ $6, 6, 2, 2, 2$ $27$ $6$ $( 1, 2)( 3,17)( 4,18)( 5,12, 8,16, 9,13)( 6,11, 7,15,10,14)$ $6, 6, 2, 2, 2$ $27$ $6$ $( 1, 2)( 3,17)( 4,18)( 5,12, 9,13, 8,16)( 6,11,10,14, 7,15)$ $3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$ $3, 3, 3, 3, 3, 3$ $6$ $3$ $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,15,14)(12,16,13)$ $3, 3, 3, 3, 3, 3$ $18$ $3$ $( 1, 5,11)( 2, 6,12)( 3, 7,13)( 4, 8,14)( 9,15,17)(10,16,18)$ $9, 9$ $18$ $9$ $( 1, 5,11, 4, 8,14,17, 9,15)( 2, 6,12, 3, 7,13,18,10,16)$ $9, 9$ $18$ $9$ $( 1, 5,11,17, 9,15, 4, 8,14)( 2, 6,12,18,10,16, 3, 7,13)$

## Group invariants

 Order: $162=2 \cdot 3^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [162, 19]
 Character table:  2 1 . . 1 1 1 1 1 . . . . . 3 4 3 3 3 3 1 1 1 4 3 2 2 2 1a 3a 3b 3c 3d 2a 6a 6b 3e 3f 3g 9a 9b 2P 1a 3a 3b 3d 3c 1a 3c 3d 3e 3f 3g 9a 9b 3P 1a 1a 1a 1a 1a 2a 2a 2a 1a 1a 1a 3e 3e 5P 1a 3a 3b 3d 3c 2a 6b 6a 3e 3f 3g 9a 9b 7P 1a 3a 3b 3c 3d 2a 6a 6b 3e 3f 3g 9a 9b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 X.3 2 2 2 2 2 . . . 2 2 -1 -1 -1 X.4 2 -1 -1 2 2 . . . 2 -1 2 -1 -1 X.5 2 -1 -1 2 2 . . . 2 -1 -1 -1 2 X.6 2 -1 -1 2 2 . . . 2 -1 -1 2 -1 X.7 3 . . A /A -1 B /B 3 . . . . X.8 3 . . /A A -1 /B B 3 . . . . X.9 3 . . A /A 1 -B -/B 3 . . . . X.10 3 . . /A A 1 -/B -B 3 . . . . X.11 6 . -3 . . . . . -3 3 . . . X.12 6 3 . . . . . . -3 -3 . . . X.13 6 -3 3 . . . . . -3 . . . . A = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 B = -E(3) = (1-Sqrt(-3))/2 = -b3